where we denote by \(\mathbb {I}\left[V^\Delta \right] (x,t,p)\)  the values of the discrete function \(V^\Delta \)  computed at \((x,t,p)\)  obtained using a standard interpolation operator. If the interpolation \(\mathbb {I}\)  is monotone, the resulting scheme is consistent, monotone and \(L^\infty \)  stable, and therefore convergent via Barles-Souganidis theorem (see [1]}). Some typical examples of monotone interpolation operators are \(\operatorname{\mathbb {P}}_1\)  (piecewise linear on triangles/tetrahedra) and \(\operatorname{\mathbb {Q}}_1\)  (piecewise multilinear on rectangles) interpolations.
